Optimal. Leaf size=90 \[ \frac{3 A \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3}}-\frac{3 (A+4 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right )}{8 b^2 d \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.0814465, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {16, 3012, 2643} \[ \frac{3 A \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3}}-\frac{3 (A+4 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right )}{8 b^2 d \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3012
Rule 2643
Rubi steps
\begin{align*} \int \frac{\left (A+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(b \cos (c+d x))^{4/3}} \, dx &=b \int \frac{A+C \cos ^2(c+d x)}{(b \cos (c+d x))^{7/3}} \, dx\\ &=\frac{3 A \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3}}+\frac{(A+4 C) \int \frac{1}{\sqrt [3]{b \cos (c+d x)}} \, dx}{4 b}\\ &=\frac{3 A \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3}}-\frac{3 (A+4 C) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{8 b^2 d \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.653806, size = 104, normalized size = 1.16 \[ \frac{6 A \tan (c+d x) \sqrt [3]{\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )}-(A+4 C) \sin \left (2 d x-2 \tan ^{-1}(\cot (c))\right ) \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{3}{2};\cos ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right )}{8 b d \sqrt [3]{b \cos (c+d x)} \sqrt [3]{\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.331, size = 0, normalized size = 0. \begin{align*} \int{ \left ( A+C \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sec \left ( dx+c \right ) \left ( b\cos \left ( dx+c \right ) \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )}{\left (b \cos \left (d x + c\right )\right )^{\frac{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{2}{3}} \sec \left (d x + c\right )}{b^{2} \cos \left (d x + c\right )^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )}{\left (b \cos \left (d x + c\right )\right )^{\frac{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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